Thanks to the properties of mollifications, it is very easy to prove Urysohn's lemma in the euclidean space with the big plus that the constructed function is smooth.
I was wondering if something of the kind could be achieved with Tietze extension theorem in the euclidean setting (i.e, can we always find smooth extensions of smooth functions in a closed set of the euclidean space?)
The proof of Tietze's theorem runs by constructing iteratively a family of functions with the aid of Urysohn's lemma. This family extends the original function and converges uniformly.
This is very nice, but I if I try to replicate the proof, I have no way to prove the uniform convergence of the derivatives, so I guess that if the theorem is true it will not follow these lines.
Thanks